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 12
Shorthand Usage Guide
Previous Section: 12 Pronouns
or Symbols in Mathematics
1 Pronouns, Labels and Place-Holders
1.2 Pronouns in Mathematics
In speaking and writing, we use pronouns. In mathematics, we use shorthand
notation. The use of pronouns and of shorthand notation is similar. They both
make communication simpler. In talking about numbers and quantities, we don't
use the personal pronouns he and she. We may use the pronoun it.
But when we speak about several quantities, we cannot use the pronoun it for
all of them. We need extra pronouns, labels or place-holders for these numbers
and quantities.
For example, recall the formula A = W·L used to
calculate the area A of a rectangle with width W and length L.
We can make a link or analogy between the use of letters (like A, L
and W) and the use of pronouns. In place of the single pronoun it, we
use three letters A, L and W. Instead of over-using the
pronoun it (we would need several) to refer to numbers and quantities, we
use these and other letters. When using the word it we must be careful.
What the word it represents in each sentence should be obvious from the
context. We must take similar care with substitutes (above A, L, and W ) used
for the word it.
More Examples: You may have seen shorthand notation in the formula Acircle
= pr2 for the area Acircle
of a circle of radius r. You have seen shorthand in the formula for the
area Atriangle = ½bh for the area of a triangle
with base length b and height h. The letters appearing in these
formulas with and without subscripts are examples of shorthand. They serve as mathematical
pronouns.
Shorthand in mathematics is often given by symbols or letters. The letters
are often taken from the Roman, German, Greek or Hebrew alphabets. The letter p
comes from the Greek alphabet. The selection (and corruption) of letters from
many alphabets increases the number of pronouns, labels and place-holders for
the numbers and quantities we mentioned.
The shorthand notation for a number or a quantity may become a temporary or
permanent pronoun for it. A name for an object provides a pronoun with a short
or long term attachment. We try to use one name or pronoun per item when writing
and speaking.
In summary, the role of shorthand notation in mathematics is like that of
pronouns in everyday speech. Instead of describing fully a number or quantity
when we refer to it in conversation or a calculation, we introduce a shorthand
symbol for it. The main guide in the use of pronouns or shorthand is clarity.
Avoid overuse and clearly say what each mathematical pronoun means. In any
situation, each pronoun and symbol should represent one and only one number or
quantity. To avoid confusion, care has to be taken so that different symbols,
our new pronouns, refer to different numbers and quantities. Plays in which an
actor plays more than one role without the audience being told can be confusing
as well - Can actors be viewed as living pronouns?
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 Section: 12 Symbol Overuse, Too Many
Roles for one letter or symbol
Next Chapter: 13 What's Next
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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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