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YOU are better than YOU think. Show
yourself how:
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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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Dependent or Independent Variables,
A Matter of Choice
The area A of a rectangle is given by the product of its dimensions, say
width W and length L. In algebraic shorthand notation
A = WL
That being said, if we choose to calculate A from the values of W and L, then
A will be called the dependent quantity or variable while W and L will be called
the independent variables or quantities. On the other hand, we choose to compute
W in the equation A = WL from the values of A and L, then W = A/L and we will
call W the dependent variable and will we call A and L the independent
variables. So which variable is dependent or not depends on how we choose to use
a formula, or which ones we choose to give or calculate. If we are situation
where we know A = WL but we do not know which will be given and which will
be given, we cannot say which one will be dependent, albeit the form A = WL
suggests A will be the dependent variable. What we can say for sure or
certain is the the equation A = WL links or relates the area, width and area of
a given rectangle. The set theoretic view or codification of a relation
will be given below.
Postscript - Another Reading Guide
Arithmetic and algebraic expressions and formulas are like pictures better
read in silence than spoken aloud. That situation has led to a lack of words
in mathematics. The Discussion of Three Skills for Algebra in Chapter
8 to 12 provide a wordy, or too wordy remedy. See how much you swallow
today, and return for the rest to perfect your skills and comprehension later.
Note too, further skills for algebra, a fourth and/or fifth appear in Chapter
14. That chapter introduces two unifying themes for secondary school
mathematics:
- The forward and backward use of almost all formulas and equations (those
in function form y = something)
- The connection between arithmetic (or numerical) solution and algebraic
solution methods in the backward use of formulas and equations. If you
understand the algebraic solution, you are halfway to understanding the full
strength use of algebra in senior high school mathematics and
calculus.
So now, read on and get a headstart or catch-up for
high school mathematics and/or calculus. Good luck.
See too Chapter 7 with its Arithmetic Skill Check
for Calculus and Pre-calculus Students - Is your arithmetic precise and
efficient?
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www.whyslopes.com
Volume 2, Three Skills for Algebra -
Preview, starter & further lessons for logic and algebra
to (i) improve work & study skills; (ii) to to ease or avoid
algebra (math) fears & difficulties; and (iii) to fill gaps in the
exposition of mathematics.
Foreword, Chapters and Appendices follow.
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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