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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Here are five more suggestions for learning well.
A Look for New Ideas
First, learning will be the most rapid when you look for ideas new to you.
They make your studying a subject for the first or last time, worthwhile.
Meeting familiar ideas may refresh your memory but you only learn from new
ideas. The familiar ones will take care of themselves. When you can find no more
ideas new to you, your study or review of a subject is done - or you are tired
and you need a rest before starting again.
B Look for Names
Second, look for the names of ideas, places and people. Coin or invent names
if need be. An idea, object or person once named or identified is yours to speak
or write about. Names may banish the vague or unclear use of the words it and
they from your speech and writings.1
1In
old and possibly some present-day religions, people thought that another's
knowledge of their names made them vulnerable to spells and curses. Today
knowledge of your name, address or various identification numbers make you
more vulnerable to a loss of privacy, junk mail, and legal proceedings -
a kind of spell different from that previously envisioned or anticipated
yesterday.
C Proceed (Go) Step by Step
Third, find a step by step explanation or description of the skills you need.
Then begin your study of a subject at a step where everything is understood.
Starting elsewhere leads to confusion. It may cause you to turn away from the
subject. In particular, if you try to master a subject too quickly, you will
find some skills are too hard. Then you will need to start again with the
simpler ones before them. You may further need to ask someone about which skills
you will need to know to master the ones hard for you. Lessons taught by others,
or self-taught, must build new skills on those previously mastered.
D Be Persistent
(Be Stubborn)
Fourth, understanding whatever you have to do or read takes time. If you have
ever done a crossword or jigsaw puzzle, then you know you can work on one part
of the puzzle and then another. Each part you do, manage to solve or understand,
may help with the rest of the puzzle. The same is true of a book or class notes.
Read them in order if you can. But do not be afraid to look ahead, or behind,
for clues to what the current passage or word means. So read with patience. Be
prepared to puzzle or think. Further, do not hesitate to get a second opinion or
view from another person or book. The words of others may provide a path to
follow but the understanding of any subject is an individual effort. No one else
can do this for you. So be persistent and look for the ideas new to you.
E Ask Why
Fifth, ask your teachers why each course or lesson is given, or what they
hope to show you. Reasons for learning this or that can be requested.
Sometimes we teachers cannot give a full answer - our
employees (School boards) may have told us to teach you a topic based on great
or little wisdom. In education, too many cooks spoil the broth.
Appendices with (repetitive) advice for Students: [ B How to Learn ] [ C. How to Read ] [ D. What to do in School ] [ PS. Study Tips ] [ PS: Time and Effort ] [ E. How to Study Math and Why ]
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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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