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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
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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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Three Notions of a Variable
What is a Variable, Sections: [ Up ] [ 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 ]
The concept of a variable is not simply described in most algebra texts. A
clarification follows. This clarification is not for the expert, but for the
novice. The specialized use of the term variable should not be the first
one given in an algebra text or dictionary, mathematical or not.
A First Notion: Variables Without Symbols. We can talk about numbers
and quantities, and among them identify those which are changing or varying, and
those which are constant, known, unknown, given, confidential and so on. Here a
number and quantity which may vary, or take many values in the circumstances of
interest, is called a variable. We can talk about variables without using
the shorthand notation, that is, letters and symbols, employed in algebra.
Examples follow below.
Second Notion: Variables with Symbols. Formulas use shorthand
notation, symbols or letters, to represent numbers and quantities. This suggests
that when a symbol or letter is the shorthand notation for a number or quantity
which may vary, we may also call that symbol or letter a variable.
Remark 1. The association of symbols and letters with
numbers and quantities which may vary is so much a taken-for-granted part of
the algebraic way of writing and thinking (amongst the mathematical adept)
that the observation that we can talk about variables apart from symbols has
been overlooked. But this symbol free notion clarifies and refines the concept
of a variable in mathematics.
Remark 2. The notion that a variable may be given
by a symbol, that is shorthand notation (or a place holder) for a number or
quantity which may change, relies on our ability or skill (i) to talk
about numbers and quantities and also on our ability or skill (ii) to employ
shorthand notation (symbols) for them in and possibly outside calculations.
Third Notion: Variables and Computer Memory
Locations: Computers and calculators may be
used to store the values of numbers, amounts and quantities in named or labeled
memory locations. Computers or calculators may be programmed
to use or change the stored values of numbers or quantities, values that may
vary. The values, the memory locations where they are stored, and the
names or labels for them may be all be called variables.
Three Skills for Algebra
- We can talk about numbers and quantities. The words or adjectives
used here may be used in mathematics after arithmetic. There is
more to mathematics than just doing arithmetic.
- We can describe calculations that might be done (or postponed) with
words alone or with an (algebraic) shorthand notation. The description of
calculations that might be done is also part of mathematics after
arithmetic. There is more to mathematics than just doing arithmetic.
- We can change the way a number or quantity is computed. Some
rule-based reason is required here. There is more to mathematics than
just doing arithmetic.
Talking about these skills and emphasizing them in examples shows there is
more to mathematics than just doing arithmetic.
Chapter subsections: [ Up ] [ 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 ]
Next: Constants
and Parameters
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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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