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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Introduction with Advice and Directions
In any art, trade or discipline, from the master the
apprentice learns rules and patterns to use and combine, carefully, with the
aim of obtaining repeatable, reproducible and verifiable results. In
arts & discipline with long histories, the rules and patterns, discovered
and refined over time, are not all obvious to the beginners. Whence a master
or teacher must show the way. Welcome to the online words of a logic and
algebra teacher-
The foreword to Volume 2 gives an overview of
book material. The online pdf
version of Volume 2 includes its foreword, chapters and appendices but not
the online postscripts and arithmetic videos, nor this page.
Success in school in large part depends on your will to
learn or willingness to sit down and study. Do so here or
elsewhere. Good luck. If site material is too hard, join or form a study
group. If site material is easy, lead one.
Logic mastery is a key for enriching skills
and understanding, and a must for easing or avoiding difficulties in school and
work, those difficulties due to imprecise reading and writing.
If you plan to study calculus, do the Arithmetic
Review Problems and have them corrected by another. Problems like these
catch errors that slow or undo too many students entering
calculus. Read chapters 1 to 14 in this Volume 2 do not require you to
review arithmetic. They can be easily understood without arithmetic
skill review and perfection. But you should put that review and
perfection, alone or with help, on your to do list.
Senior High School and College Students: Do the
arithmetic review problems now or later to improve your arithmetic skills and
see numerical hints or support for the algebraic way of writing and
reasoning. See the webvideos for arithmetic (not all at one please) to
review and consolidate numbers skills if not sense you should have obtained in
junior high school.
Altogether, the logic chapters provide a unique mathematics-free
introduction to the direct and indirect definition and rule-based thinking that
appeared in Euclid's work a long time ago (2300 years ago)
- While some rules are learnt by rote (without comprehension
of why they work), seeing how they combine to produce further rules and
practices starts a thought-based development of skills and concepts. Good
Luck.
- What is hard for a 10 year old is easier for a 14 year old,
and easier still, we hope, for older learners. So if area lessons on logic
are too hard return in a few years, or return with another student or
teacher to unravel any mysteries. Two minds may be better than
one. Good luck. Sit down and study here and elsewhere.
Words have been missing or misused in earlier
introductions of algebra possibly because arithmetic & algebraic
expressions are better read and understood silently than read
aloud. Online chapters and postscripts enlarge and clarify the role of
words in elementary mathematics.
Algebra is hard for many. By learning to
talk about number and quantities, describe them with words, before and then
besides the shorthand roles of letters and symbols, we can more clearly
understand and explain the algebraic or mathematical way of writing and
reasoning. See the long essay What is
a variable and the Algebra chapters 8 to
12. Here are some easy lessons and technical details that should be met
and mastered sooner or later, with sooner better.
- Formulas Forwards and Backwards: Every formula
you meet in high school and college will be used backwards and forwards. The
arithmetic approach to this may be easiest for you in the first instance,
but if you master the algebraic approach, you will have stronger base and
stronger abilities in mathematics for further studies. Chapter 14 employs
the Compound
Interest formula directly and indirectly (forwards and backwards), and
compares arithmetic and algebraic ways for this.
The backward or indirect use of formula represents a fourth skill for
algebra or a refinement of the third. Should Volume 2 be renamed Four
Skills for Algebra?
- Solving linear equations in one or two unknowns is another key
part of junior and high school mathematics. Explore the site area Solving
Linear Equations with Stick Diagrams for an introduction and then come
back to Chapter 15 in this Volume. If you can
solve two equations in two unknowns, and do arithmetic with fractions
exactly and efficiently without the aid of a calculator when the numerators
and denominators are say < 101, you will have good base for senior high
school mathematics and senior high school courses in science and accounting.
- Chapter 18 Rules of Algebra describes how to use
or interpret the algebraically described axioms (patterns) of real number
arithmetic, one at a time and one after another. It is a long chapter. I
plan to rewrite it to add more details or make existing explanations
clearer. To learn a subject well, to perfect skills and knowledge, there is
no choice but to master this material sooner or later. The choice is yours.
Earlier is better. New site postscripts on Geometric
origins of the distributive law, a partial explanation for positive
numbers, and the geometric introduction of how to multiply polynomials,
another partial explanation, may learn operations that work not only for
positive real numbers but for all real numbers as well.
Teachers: Here some algebra
lesson plans for the more effective development of the shorthand ways of
writing and reasoning.
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www.whyslopes.com
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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