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 |
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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 10
Describing & Changing Calculations
Previous Section: 10 Shorthand Notation (formulas) for calculations
2 Changing Calculations
The compact description of formulas using shorthand notation is useful for changing the way calculations are done. Note that when two calculations give the same result, one can be done or written instead of the other. This is the replacement principle. The rules of algebra (more precisely rules which say when two different calculations give the same result) tell us when one calculation can be replaced by another. These rules, to be seen later, are also stated or described with shorthand notation.
2.1 First Box Volume Formula
The volume of a box is given by the height times the width times the length of the box in question. More precisely,
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To begin our next line of reasoning, we will group the multiplication as follows.
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In shorthand notation, the volume V of a box is given by
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The base of the box is a rectangle with area A = L·W.
This gives
The symbol A and the product L ·W both represent the
area of a rectangle. Here A gives the result of computing the product L
·W. The product tells us the value of A. So in describing the
volume calculation, we can replace the symbol A by the product W·L,
or vice-versa, as convenient.
Our second and alternate formula for the box volume is V = H·A
where A represents the base area. Suppose you met someone who accepted
this alternate formula but who doubted our original formula for the volume. What
can we do to convince him or her that our original formula says how to compute
the volume as well? The following words may help.
To convince the person, we first recall and try to use the base area formula A
= L ·W. Let's hope this is accepted. Now if some one gives us the
width and length of the base, we can calculate from the rectangle area formula A
= (L ·W) and then compute V using the equality V = H·A
. This suggests that the original calculation V = H·(L·W)
for the volume of the box because the single symbol A and the computation
L·W both represent and both can be viewed as shorthand for the
same quantity, namely the area of the base. So the symbol A and
expression W ·L can each replace the other, whether or not the
values of A, L and W are known or not.
In closing, this suggests, we can go back and forth between these two ways of
computing the volume of the box. We can use whatever is the most convenient -
requires the least amount of work.
2.2 Second Formula
where the letter A is our shorthand for the result of the product L
·W. But expression L ·W equals the area of the base.
Therefore, an alternate formula for the volume is V = H·A
where A stands for the result of L ·W or the area of the
base. The alternate formula can be used if the dimensions L and W
are given or measured. The alternate formula can also be used if the base area A
is given, but the values of W and L are unknown (or forgotten).
But whether unknown, known or forgotten, their product L·W must
equal the area A.
V = H ·A
2.3 Back to the First Formula
Chapter Sections:
Next Section: 10. Replacement Principle
Next Chapter: 11 Why Shorthand
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
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 lessonSolving 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