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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Chapter 9
Talking about Numbers or Quantities
Chapter Sections: [ Up ] [ 9 Numbers & Quantities ] [ 9 Everyday Words ] [ 9 Words Math Usage ] [ 9 Precision or Not ] [ 9 Numbers & Quantities ] [ 9 Changing Units ] [ 9 Further Readings ]
3 Mathematical Usage of Words
The above examples show how everyday words are used to describe numbers and
quantities. Our next task is to say further or more precisely how the words
variable, constant, known and unknown are used to both describe and refer to
numbers and quantities.
See the postscript: What is a
Variable
A. Variables Versus Constants
To say that a number or quantity is variable means that the number or
quantity may vary or change. To say that a number or quantity is constant means
that its value remains unchanged. For example:
- The letter p »
3.14159... stands for or denotes a constant - a value or number which will
never change.
- The time of day is always changing. So time is varying. It is an example
of a number or quantity which is always increasing and therefore variable.
When you ask what time it is, you will get an approximate answer.
To complicate matters further, numbers and quantities may change in one
period and not in another. The height of house increases slowly as it built,
remains constant while it used, and decreases rapidly if it torn down. So this
height may be variable in some situations and constant in others. In everyday
life and in mathematics, when a number or quantity is called a constant, we
expect its value not to change in the situation at hand. Similarly, when a
number or quantity is called a variable, we should expect or suspect that its
value may change.
More examples: Your height is a variable or it was a variable while
you were growing. The speed of a car or a bicycle is an example of a variable (a
variable number or quantity that is). The speed of a car can be almost constant.
The zero speed of a stationary car or a parked car is constant - in one
reference system at least. Note that a number or quantity can be variable in one
situation, and constant in another. We can further talk about a previously
constant or a previously variable number or quantity.
In summary, the terms constant and variable can be used to talk about and
describe numbers and quantities. A constant is a number or quantity whose value
is expected not to change - whose value should not change. A variable is a
number or quantity whose value does or might change. The use of these terms is
flexible and context dependent. What is constant in one situation may be varying
or changing in another.5
5In
some algebra texts and in some dictionaries, the term variable means or refers
to the letters that appear in formulas. That use of the term variable departs
from the use and meaning given above. In my view, the mathematical usage of
everyday words should be in the first instance linked and extracted from their
ordinary usage. Where the mathematical usage has departed from the everyday
usage, we need to ask if that departure is necessary, and whether or not the
departure should be corrected. Documenting reasons or possibly causes
for such departures could be material for a thesis in linguistics.
B. Known Versus Unknown
Numbers and quantities can be known or unknown. You may know your own height,
age and weight, but I don't know your personal measurements. To you these
quantities are known. To me they are unknown. Whether they are known or not
depends on the company you keep - that is to whom you speak. When you see the
instruction find the unknown, you should ask the question: unknown to
whom? Note further in solving an equation, the solution of the equation goes
from being unknown to being known.
This is a note mainly for people who know how to solve
equations. See
the following chapter or chapters to learn how.
There is only one number x solving the equation
2x = 10. Before you solve this equation, its solution, the number
x is unknown to you. The solution is x = 5. When or as you
solve the equation (or see the solution), the number x becomes
known.
When you are only speaking about the solution x
of the equation 2x = 10, the solution is given by a constant. The
letter x stands for the constant, non-changing number 5.
Now in two different problems in which you solve for x,
their solutions x are often given by different numbers
(constants). Thus the value of the solution x may change as you
go from one problem to another. From this perspective, the solution x
can be also called a variable.
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For the sake of variety in our speech, numbers and quantities are also called
parameters. A parameter is another name for a number or a quantity. When we say
a number or quantity is a parameter, we have no immediate expectation that the
number or quantity in question will be constant nor that it will be variable.
The term parameter gives a vague expectation somewhere between constant and
variable. We can talk about numbers and quantities in precise and imprecise
ways.
Chapter Sections: [ Up ] [ 9 Numbers & Quantities ] [ 9 Everyday Words ] [ 9 Words Math Usage ] [ 9 Precision or Not ] [ 9 Numbers & Quantities ] [ 9 Changing Units ] [ 9 Further Readings ]
Next Section: 9-Talking about Numbers or
Quantities, Approximate Knowledge, Precision or not.
Next Topic: What
is a Variable:
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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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