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a calculus and preparation for calculus website, etc.

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1A. Pattern Based Reason 
1B. Math Curriculum Notes
2. Three Skills for Algebra
3. Why Slopes & More Math

Mathematics Course Designers: LAMP offers food for thought.
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||Définition d'une variable || Algèbre || Arithmetique || Logique ||La raison basée sur les règles et modelés||
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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.


Chapter 2. Implication Rules

Previous Section: Second Logic Puzzle

One Versus Two Way Implications

The two puzzles give examples of implication rules. The first puzzle gives a one-way implication rule, while the second gives a two-way implication rule. The following words should further help you to see the difference between one- and two-way implication rules. Seeing this difference may help you understand better the answers to the above questions. They may also help you answer the five questions again using the two-way implication rule.

  1. A one-way implication rule says that when a first situation occurs, so must a second. It does not say that when the second occurs so must the first. The second situation may occur without the first.
  2. A two-way implication rule says that
    1. when a first situation occurs, so must a second, and
    2. when the second situation occurs, so must the first.

    A two-way rule says that when each situation occurs, so must the other. Therefore if the two-way rule is to be obeyed, when one situation does not occur, neither can the other. 

Seeing or recognizing the difference between one- and two-way implication rules makes you a more careful thinker.

One- and two-way rules, recognized or not, are what we use to reach conclusions or make judgments. One and two-way rules can be used to suggest or persuade us of what needs to be done or avoided.

Talking About Logic

As suggested above, you can give people the above rules or similar ones before asking five questions. Before you do this, you should wait for a receptive mood, especially if you are not in a classroom. For the sake of an argument and some fun, you may ask after getting an answer, are you sure? Or you may pretend a correct answer is wrong. Of course, you will admit this ruse later, and explain why you really agree (or disagree) with the answers. The aim is to see how people reason and more importantly to strengthen their thinking skills.

Logic is supposed to give rules for thought, that is rules for arriving at conclusions. Yet the only rule needed in the reasoning shown above is as follows: Read exactly what is written and don't assume nor imagine too much.

Implications Versus Suggestions

In a dictionary you may find that the verb to imply also means to suggest. Words which say when one event occurs so does or will a second are called suggestions or implications. Suggestions and implications can be true. True here means obeyed or at least not disobeyed. Suggestions and implications can be false. False here means disobeyed. In our reasoning process, we want to say with certainty that when this occurs so will that. In practice, we may have to be content with saying when this occurs, so may that. Knowing which of our rules are sure or which are uncertain identifies the weaknesses in our reasoning processes. The implication rules that are never disobeyed provide the most certain suggestions in reason.

In logic, when we speak of implication rules, we speak of rules which we hope are never disobeyed. Rules which might be disobeyed are called conjectures, suggestions or guesses. Evidence (persuasion) should be required to convince us that a conjecture or suggestion is a reliable implication. We can imagine or suggest more than we can prove. Caution is advised on hearing a rule. Before applying a rule, you need to know how certain it is. Is it a reliable implication or merely an uncertain suggestion?

Implications Versus Suggestions

In a dictionary you may find that the verb to imply also means to suggest. Words which say when one event occurs so does or will a second are called suggestions or implications. Suggestions and implications can be true. True here means obeyed or at least not disobeyed. Suggestions and implications can be false. False here means disobeyed. In our reasoning process, we want to say with certainty that when this occurs so will that. In practice, we may have to be content with saying when this occurs, so may that. Knowing which of our rules are sure or which are uncertain identifies the weaknesses in our reasoning processes. The implication rules that are never disobeyed provide the most certain suggestions in reason.

In  logic, when we speak of implication rules, we speak of rules which we hope are never disobeyed. Rules which might be disobeyed are called conjectures, suggestions or guesses. Evidence (persuasion) should be required to convince us that a conjecture or suggestion is a reliable implication. We can imagine or suggest more than we can prove. Caution is advised on hearing a rule. Before applying a rule, you need to know how certain it is. Is it a reliable implication or merely an uncertain suggestion?

 

Chapter Sections: Up ] First Logic Puzzle ] Second Logic Puzzle ] [ One-versus Two-Way Implications ] Implications versus Suggestions ]

Next Section:  Implications versus Suggestions

Next Chapter: Chains of Reason - Euclidean Model for Reason

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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

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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