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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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.

 Chapter 11
Why Shorthand

Previous Chapter: 10 Describing & Changing Calculations

1  Words are Not Enough

Imagine describing a picture by saying there is a patch of blue, a bit of red or orange there and yellow here. Words alone do not usually describe a picture fully. The picture must be seen. A picture seen conveys or presents information which words alone cannot capture. Similarly, formulas and arithmetic calculations are better seen than described with words alone. Pictures and formulas are often worth more and may be easier to understand than a thousand words. Words cannot replace pictures and formulas.

2  Decimal Notation in Arithmetic

Decimal notation is used in the representation of whole numbers, fractions and irrational numbers. Decimal notation allows us to write down these numbers easily. Further, before decimal notation appeared, addition, multiplication, division and subtraction were complicated operations. People had to use the abacus or Roman numerals or other means for calculating. Imagine, if you can, how to do arithmetic without decimal notation. Decimal notation is the key to our doing arithmetic easily. Decimal notation is itself a shorthand notation for the representation of numbers.

Since the 16-th century A.D. popularization of decimal notation, rules and methods have been invented or found to say how to write, add, multiply, divide and subtract numbers and fractions with this notation. Today these methods or variants of them are taught in elementary school.¹³

Good or understandable shorthand helps people to describe calculations and even to do them. The ease with which we write or record numbers and calculations affect the way in which calculations are done and if they are done. Before the use of decimal notation, basic arithmetic operations were difficult to do and record, few people mastered arithmetic.

In talking and writing, our language and vocabulary and experience limit and guide what we can say or even think of. Similarly, in mathematics the language (shorthand conventions) and previous experience limit and guide what can be said or attempted.

3  Shorthand Notation in Algebra

The use of shorthand notation (letters or symbols) in the description of calculations is only several centuries old in Europe. Prior to the 12-th century A.D., calculations were described with words alone. The use of shorthand notation for describing calculations is not old. In geometry letters have been used by some Greeks, two thousand years ago, to label the sides or vertices of triangles, but not to describe calculations.

As in arithmetic, algebraic shorthand notation records and describes calculations in a written, visual and non-verbal manner. Here words are not enough to describe simple calculations. For instance think how you would describe the calculation of the area of a rectangle or a triangle to a young child with words alone. But in the description of more complicated calculations, words become inconvenient and awkward to use. The description is best given with the so-called algebraic shorthand notation of mathematics. For examples of more complicated calculations, consider the compound interest formula and the quadratic formula.

 

1. The compound interest formula:

   A = P(1+i)n

2. The quadratic formula:

   x = -b ±   ______ Öb2 -4ac 2a

Complicated formulas are written to be seen. They are not easily read aloud. Their meanings are not easily described without the use of shorthand notation. The shorthand notation in mathematics gives a written code in which calculations are efficiently and briefly described and/or changed. It is a code which is better seen than read aloud.¹⁴ The algebraic shorthand provides a language and environment in which calculations are changed or manipulated into new ones.

4  Seeing is Better Than Hearing

The saying seeing is better than hearing usually means that seeing and witnessing an event is better or more reliable than hearing about it. In this section, we will give a new meaning to this saying.

Our ability to see is more powerful than our ability to hear. Words are typically understood one at a time. After that they need to be remembered. In contrast, a picture, a calculation or a formula can be seen all at once. They will remain in front of you for further thought and observation while you are looking. No memory is required. Once a picture, formula or calculation has been put on paper, the details can be seen all at once. The paper used this way, serves as an immediate or quick extension of our minds or memory.

Eyesight together with algebraic shorthand notation and geometric diagrams provide more powerful ways for the communication and recording of mathematical thoughts than words alone. All this explains why mathematics is better seen and written than spoken aloud. Words alone are not enough for the communication of mathematics.

Footnotes:

¹³The development of arithmetic during the period 1300 to 1637 A.D. is described in Chapter XI of the book A Short Account of the History of Mathematics written by W. W. Rouse Ball (Publisher Dover, New York 1960). A copy should be in your town or school library. The book was written in the last part of the 19-th Century.

The book A Source Book in Mathematics by David Eugene Smith, first printing 1929, Dover reprint 1959, has a chapter on the 1585 A.D contribution of Simon Stevin (1548-1620) to the popularization of decimal notation. More recent accounts of the history of arithmetic and decimal notation may modify or correct the historical impression in the above references.

¹⁴We need a code which could be read aloud more easily.

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

Real Player Videos Perfect arithmetic skills with whole numbers & fractions after or besides chapters 1 to 14. Arithmetic Videos Summary Addition with Decimals Subtraction with Decimals Multiplication with Decimals Fraction Arithmetic Recognizing Primes Long Division for Decimals Square Root Simplification Greatest Common Divisors Least Common Multiples

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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