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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Refer to methods for obtaining Prime Factorization, GCDs, LCDs as
needed in the other video pages.
- [Play
Video] 3-4 minutes. Equivalent fractions - Lowering and
raising terms (the values of numerators and denominators) to
obtain equivalent fractions. Simplification involves lowering
terms - cancelling common factors or divisors on top and bottom.
Addition & subtraction of fractions may involve raising terms
to obtain a common denominators. See below.
- [Play
Video] 2-3 minutes A few examples of Simplifying Fractions -
lowering terms by canceling common factors until there are no more
common factors, so that the numerator and denominator are
relatively prime, that is there prime decompositions have no
primes in common.
- [Play
Video] 2-3 minutes. Multiplying Fractions with
cancellation of common factors done first (recommended) or
not, with more simplification to be done later.
- [Play
Video] 5 minutes. How to add fractions using common
denominators. Here the common dominators is the lowest or
least common denominator (LCD) and its given by the least common
multiple (LCM) of the denominators in the fractions added
together. Here the listing multiples method is used
to compute the LCM. The alternative of not using the LCD for the
fractions is explored to show what happens when the LCD is not
used.
- [Play
Video] 3 minutes Another example of how to add
fractions with and without the least common denominators with
an explanation that not using the LCD (least common
denominator) leads to ratios that can be simplified. So use
of LCDs is promoted.
- [Play
Video] 3 minutes - Comparison of Fractions Size or
Magnitude, and more examples of the use of common denominators
in addition and subtraction.
- [Play
Video] 3 minutes - Another example of the listing multiples
method to find the LCM and thus the LCD for the sum of two
fractions.
- [Play
Video] 4 minutes - Factorization method to obtain
a common denominator, here the LCM and thus the LCD for the sum of
two fractions. See if you can recognize the GCD of the
denominators here. It is not mentioned here. In this
example, the LCD is given by a product that does not have to
be evaluated explicity due to cancellation of common terms after
addition of fractions.
- [Play
Video] 2 minutes - Fraction Simplification using Prime
Decomposition (factorization) to identify common factors
for cancellations.
- [Play
Video] 5 minutes - Product Simplification using Prime
Decomposition by Canceling Common Primes, thus avoiding some
denominator and numerator multiplication. An alternative common
factors as they appear, more opportunistic, is given and is to be
recommended.
- [Play
Video] 5 minutes - How to use Prime Factorization or
Decomposition for LCM and LCD for a pair of denominators, an
example.
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The simplification, multiplication and addition of Fractions may depend on
recognition and cancellation of common factors, prime or not. See how GCDs and
LCMs (or LCDs) may be used in the addition and multiplication of fractions.
Euclid's Algorithm for computing the GCD of a
pair of whole numbers provides a method for simplifying fractions, quickly
without using prime decomposition of numerators and denominators.How should
appear in a future video.
More Videos: [ Up ] [ 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 ]
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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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