Appetizers and Lessons for Mathematics and Reason (www.whyslopes.com)
||Définition d'une variable || Algèbre || Arithmetique || Logique ||La raison basée sur les règles et modelés||

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1A. Pattern Based Reason 
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3. Why Slopes & More Math

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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.


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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.

The Second Puzzle

Previous: 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.

 
  1. When the rule is obeyed, what can you say happens for sure when Aunt Jane visits her nephew's home? This is easy.  [Answer
  2. When the rule is not disobeyed, what can you say happens for sure about Aunt Jane when Tom is out playing? Be careful. [Answer]
  3. 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]
  4. What must happen for the given rule to be disobeyed? This is another easy question. [Answer]
  5. 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

Answers to the Second Puzzle

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.
  1. 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).
  2. 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).
  3. 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).
  4. 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).
  5. 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 Puzzle ] [ Second Puzzle ] One- Versus Two-Way ] Talking About Logic ] Implications vs Suggestions ] One Versus Two Way Committments ] Repeatable & Reproducible ] Limits and Benefits ] Accidental Rules ] Steps for Better Reason ]

Next: One- Versus Two-Way Implications

 

www.whyslopes.com
Volume 1A, Pattern Based Reason

 Chapters 1 to 24

FOREWORD
Three Remarks

1 Introduction
2 Communication
3. Elements of Reason
4 Implication Rules
5. Deception
6 Chains of Reason
7 Longer Chains
For & From Consistency
8. Language Change
9 Next Chapters
10 Responsibility
11 Accidental Patterns
12 Knowledge Islands
13 Euclidean Logic
14 Deductive & Empirical Views of Mathematics
15 Objectivity
16 Origin of Rules
and Patterns
17 Objective Ways

18. Waking up
19. Symbols  & Logic
20. Pronouns or Symbols
21. Truth Tables I.
22. Truth Tables II
22. Biconditional
22. Contrapositive
23. IF-THEN table
24. Indirect Reason Again

To reason often means to persuade someone of the need for an idea or action. That someone could be yourself. So be careful.

1A Logic Postscripts
- online only

+Proof by Absurdity alias proof by contradiction
+How the demand for consistency supports the law of the excluded middle
+Reality versus or with the aid of Imagination
+Links for reason, logic and crtical thinking
+Three Remarks
+History Lost or Missing

There is a difference between
knowing how to spend money,
and having money to spend.

There is likewise a difference
between mastering a skill
and having meeting a situation in which it applies.

 



 


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