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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 2,
Implication Rules
The Second Puzzle
Previous Section: The First
Puzzle
A Two-Way Implication Rule
Try answering the five questions again, using this two-way (implication) rule
Tom goes out to play when and only when Aunt Jane visits his home.
instead of the original rule. How will the answers change? Rather, which
answers change? This second rule can be restated as follows.
Tom goes out to play when Aunt Jane visits his home.
and also
Tom goes out to play only when Aunt Jane visits his home.
The first when part of this rule is disobeyed in the situation where
Aunt Jane visits and Tom does not go out to play. The only when part of this
rule is disobeyed in the situation when Tom goes out to play without his Aunt
Jane visiting. Here are the five questions again.
- When the rule is obeyed, what can you say happens for sure when Aunt Jane
visits her nephew's home? This is easy. [Answer]
- When the rule is not disobeyed, what can you say happens for sure about
Aunt Jane when Tom is out playing? Be careful. [Answer]
- When the rule is not disobeyed, what can you say happens for sure about
Tom when Aunt Jane is not visiting? Be careful, again. [Answer]
- What must happen for the given rule to be disobeyed? This is another
easy question. [Answer]
- When the rule is not disobeyed, what can you say for sure about Aunt Jane
when Tom does not go out to play? See the answer to the fourth question. [Answer]
Answers are given twice
- in popup boxes, and
- in text below. (as in
the printed version)
See if you agree with them.2 |
The two-way implication rule for the second puzzle is:
Tom goes out to play when and only when Aunt Jane visits his home
instead of the original rule. How will the answers change? Rather, which answers
change? This second rule can be restated as follows.
Tom goes out to play when Aunt Jane visits his home
and also
Tom goes out to play only when Aunt Jane visits his home.
The first when part of this rule is disobeyed in the
situation where Aunt Jane visits and Tom does not go out to play. The only
when part of this rule is disobeyed in the situation when Tom goes out to
play without his Aunt Jane visiting. The questions and answers follow.
- When the rule is obeyed, what can you say happens for sure when Aunt Jane
visits her nephew's home? Answer: Tom must be out playing (no change).
- When the rule is not disobeyed, what can you say happens for sure about
Aunt Jane when Tom is out playing? Answer: Aunt Jane must be visiting (the
answer has changed).
- When the rule is not disobeyed, what can you say happens for sure about
Tom when Aunt Jane is not visiting? Answer: Tom is not outside playing (the
answer has changed).
- What must happen for the given rule to be disobeyed? Answer: Either Aunt
Jane must be visiting and Tom does not go out to play or Tom must be out
playing without Aunt Jane visiting (the answer has changed).
- When the rule is not disobeyed, what can you say happens for sure about
Aunt Jane when Tom does not go out to play? Answer: Aunt Jane is not
visiting (no change).
Chapter Subsections: [ First Logic Puzzle ] [ Second Logic Puzzle ] [ One-versus Two-Way Implications ] [ Implications versus Suggestions ]
Next: One- Versus
Two-Way Implications
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www.whyslopes.com
Vol. 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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