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 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.
What is a Variable?
©Alan Selby, August 2000.
Goal: Master the mathematical use of the word variable
Previous: Chapter 9, How to Talk or Describe Numbers and Quantities
What is a Variable, Sections: [Introduction]
[ Variation between Examples ] [ Variation of Letters ] [ When does a letter denote a variable ] [ Cases of Double Variation ] [ Three Notions of a Variable ] [ Constants ] [ Talking about numbers ] [ Dependent or Independent Variables ]
Introduction
Look in a dictionary, encyclopedia and a mathematics text for a definition of what is a variable, an introduction that is understandable to you and easily explained to others. If you find such a definition or introduction clear enough to help in mathematics after arithmetic, the rest of this essay need not be read.
Alice in Wonderland if she could speak today, would observe that the view of a variable as a function begs the question of how to explain the notion of a function without using the concept of a variable. The essay or chapter before put the concepts of what is a variable first and before the use of symbols and notation in mathematics for numbers, amounts, quantities and functions.
Variation in a Single Example
variation = amount of change
The next diagram shows the height of a bird during its journey from one tree to another. The flight is over the ground intervals
[a,b], [b,c], [c,d], [d,e], [e,f]
Flight of a Bird
Letters on horizontal axis end ground intervals where the height behavior changes. If height is measured above or below sea level, and the tops of both trees were below sea level, then increasing height would correspond to make the height relative to sea level less negative.
Identify the intervals where the height of the bird is constant, where this height is increasing (becoming more positive or less negative) and where this height is decreasing (becoming less positive or more negative). The height may have different behaviors on different ground or time intervals. This exercise could be redone on a graph of height versus time. In this case, the ground intervals would correspond to time intervals.
To vary means to change. Identify the ground intervals where the height of the bird is constant (not variable) and where it is variable.
Conclusion:Whether or not a number or quantity is constant or not, variable may depend on the interval in which is observed or examined or remembered. We can talk about numbers and quantities being variable without or before the use of letters to represent them.
The following diagram shows the speed of a car along a straight road.
Piecewise linear graph of speed versus time
Identify the time intervals where the speed of the car is constant and where it is variable.
Challenge (a hard exercise):From the above diagram, how would you find the distance traveled by the car in a constant-speed interval and in the variable speed intervals. Find a solution without the use of calculus. Hint: See an old high school physic text.
What is a Variable Sections:
Next Section: Variation between Examples or Chapter 10, Describing & Changing Calculations
This webpage and its sections on What is a Variable (not part of Three Skills for Algebra) is a postscript originally posted online in August 2000. There is a hidden curriculum in mathematics in which talking about numbers and quantities, the first skill for algebra at this website, is not discussed.
www.whyslopes.com
2. Three Skills for AlgebraForeword, 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
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