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YOU are better than YOU think. Show
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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
Do not leave here without it - Logic
mastery will develops critical thinking, improve reading and
writing, and give a firmer base for work and studies at many levels.
Good luck.
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Caution: Site advice
is approximately correct, for some circumstances, not all.
Site How-TOs are
logically developed, but not tried and tested. That leaves
room for thought and refinement.. |
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After logic,
(a) continue reading Three
Skills for Algebra, chapters 8 to 14 and do so alongside
site area on solving
linear2007 Equations ; or (b) see this calculus
starter lesson and Volume 3, Why
Slopes & More Math, chapters 2 to 6;
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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Explore collaborative whiteboards
from groupboard,
twiddla or
scriblink.
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Chapter 2. Implication Rules
Previous: Introduction to Logic Chapters
Chapter sections: [chapter intro] [ First Logic Puzzle ] [ Second Logic Puzzle ] [ One-versus Two-Way Implications ] [ Implications versus Suggestions ]
In this chapter, you will meet two puzzles which show the difference between
one- and two-way implication rules. Mastering the difference is a simple, first
step, in rule and pattern-based thought. This first step is needed to precisely
read rules, definitions and statements in all disciplines, including
mathematics.
Are you a careful thinker? Can you understand exactly
the meaning of a rule or pattern? Instructions for building or creating
provide rules and patterns which say and suggest that when this is done, that
should happen. Every cook and dressmaker knows the importance of following
instructions carefully. Instructions and suggestions which are not repeatable
and results which are not reproducible are not of interest to a cook or
dressmaker.
To read rules carefully, do not imagine too much. To decide or choose among
opinions and actions, you must understand the exact meaning of written and
spoken words. You need this skill to understand, to follow, to write and to
change rules, guidelines, instructions and laws, etc.
Use your imagination in language courses. Use your imagination when you are
reading novels (and newspaper opinion columns). When reading newspapers or
listening to radio and television ask: Is the story presented in a one-sided
way? Headlines may suggest conclusions which are not in the stories or the
text. Look at the details. Here imagination allows you to guess what the full
story might be. But imagination provides only suggestion, not proof. Confidence
in suggestions must come after proof is given, not before.
Also use your imagination for poorly written rules to guess their meanings.
Guesses and speculations give possible meanings. These may or may not be
correct. Proof and evidence, or tests, may decide which among various
possibilities, if any, are correct.
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.
Chapter sections: [ First Logic Puzzle ] [ Second Logic Puzzle ] [ One-versus Two-Way Implications ] [ Implications versus Suggestions ]
Next Section: First
Logic Puzzle
Next Chapter: 3. Chains of Reason -
Euclidean Model for Reason
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[ Back ] [ Next ]
Three Skills for Algebra
www.whyslopes.com
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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