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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
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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 1
Elements of Reason
Previous: Foreword
To reason with someone often means to persuade them of the need for an idea
or action. Persuasion and reason can take many forms. Methods for arriving at
conclusions and judgments in all skills disciplines are, or should be where
possible, based on the use and recognition of reliable rules and patterns, and
also data to use with them.
Each of us needs to understand fully or as much as is
possible, whatever we might be doing or learning. In reasoning, some rules and
patterns are reliable. Others are guidelines. Each of us needs to know which
is which.
Mathematics, in and besides arithmetic, depends on rules and patterns,
which are used one at time or one after another, to obtain conclusions. The aim
of the next chapters is to introduce and provide an image of rule and pattern
thought in mathematics and other disciplines.
When ideas in mathematics or another discipline are described instead of
being drawn from implication rules, the role of rule-based reason or logic may
be forgotten. But in every discipline including mathematics, signs of rule-
and pattern-based reason are given by the word and phrases such as from
this, then, if, therefore, thus, because, since, as, gives, yields
etc.
About the Next Chapters
The next chapters describes some basic elements of rule- and pattern-based
thought. In particular, four chapters, Implication Rules,
Chains of Reason, Longer Chains of
Reason and Islands and Divisions of Knowledge
describe basic ideas, keys for reason and logic.
- The chapter Implication Rules presents two logic
puzzles. Each consisting of a rule and five questions. Answers are also
provided. The puzzles show the difference between one- and two-way
implication rules.
- The chapter Chains of Reason describes how
to directly use rules one at-a-time or chained together, one-after another,
for arriving at conclusions and judgments.
- The chapter Longer Chains of Reason starts
to indicate the special role of reason in mathematics. It describes, in a
very non-mathematical fashion, the concept of induction, a method used in
mathematics to arrive at conclusions. This concept of induction is an
example of a method of reason employed mainly or only in mathematical
subjects.
- The chapter Islands and Divisions of Knowledge describes
how rule and pattern-based bodies of thought may be organized. Here
different starting points, first principles or assumptions, may lead to the
same body of rule-based knowledge.
In philosophy, the discipline that is
literally the love of knowledge, perhaps an infatuation, Euclid's logical or
rule based arrangement of geometry provided a model for reason. This chapter
with words and images apart from geometry describes the model and the
variations possibly within it.
These chapters develop thinking and reasoning skills needed in daily life. They
provide a standard or model for arriving at conclusions and making decisions:
how to argue politely if you must. They also strengthen basic skills needed in
mathematics, science, technology, writing, persuasion and communication. Reason
and persuasion touch all skills and all disciplines.
The chapter A Change of Language introduces
the conventional if-then and iff forms for writing one- and
two-way implication rules. The one- and two-way implication rules in this work
have been identified with condition and bi-conditional statements. But the
terminology one and two-way employed here draws on the present-day common
experience of one and two-way roads.
Next: Chapter 2, Implication Rules
To Learn More About Logic, see chapters 26 to 31
in this Volume, or the identical (or supposedly identical) Chapters
19 to 24 in Volume 1A,. Pattern Based Reason .
Volume 1A describes the benefits, origins of rule and pattern based thought,
deeds and hopes in greater detail, and still leaves room for thought.
Online postscripts in the Volume 1A site area discuss further the methods and
context for indirect reason in and outside of mathematics. Finally, section C in
the webpage Mathematics
Education in General includes a theory of knowledge based on our ability to
collect, invent and tell stories with words and symbols, written, spoken or
drawn. That theory may move to its own page at some future point in time.
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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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