|
YOU are better than YOU think. Show
yourself how:
|
// _ _ \\
/\ /\
<| (o) (o) |>
\ | | /
-/[]\-
||
/ \_
||||||||||||||||||||||||||||
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;
|
// _ _ \\
/\ /\
<| (o) (o) |>
| |
| |
\
/
\ = /
|
Caution: Site advice is approximately
correct, for some circumstances, not all. That leaves room for thought |
-/[]\-
||
_ / \
||||||||||||||||||||||||||||
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 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 ]
2 Using Everyday Words
Our next aim is to show how everyday words should be used in mathematics to
describe numbers and quantities - their use here is close to their everyday
meanings. For example, we can say if a number or quantity is known or not,
changing or not, constant or not, increasing, decreasing, shrinking, growing,
confidential or embarrassing, top-secret or simply forgotten. Everyday words
give the descriptive vocabulary of mathematics. Describing and talking about
quantities and numbers is a part of mathematics after arithmetic. More examples
follow.
2.1 Airplanes or Jets
We can speak about the height of an airplane above the ground. We can speak
about it without measuring it and without knowing it exactly. The height will be
zero when the airplane is on the ground. This height increases as the plane
takes off. The height will then remain almost unchanged and nearly constant when
the plane has reached its maximum height or cruising altitude. Then at the end
of the trip, the height of the plane will decrease (get smaller) until the
plane, we hope, gently lands.
2.2 People
We can also speak about the number of people in a room. When nobody enters or
leaves, this number remains constant. When somebody enters or leaves, this
number varies. This number or count is usually a whole number or zero. When
someone is just leaving and partly in and partly out of this room, we cannot
count or we have to allow fractions.
When we speak about the number of people in a room do we mean completely in,
do we include fractions, or do we just say the count cannot be done at those
moments when someone is partly in or out, moving or not? This number or count
needs to be clearly defined. Words are needed to say precisely how it is
computed, otherwise ambiguity results.
2.3 Height
When a building is being constructed, its height is increasing. The
construction and the increase in height of the building may take place over one
or two years. While the building is used, say seventy years, its height may be
constant - unchanging. At the end of the building's useful life, the building is
left to fall down or it is demolished - torn down. Here over a long or short
time, the height decreases.
The height of the building varies. This height is therefore a variable during
the construction and the demolition (collapse or falling down) of the building.
The height is usually a constant, unchanging and invariable quantity during the
seventy or so years that the building is used.
The height of the building may or may not be known to us during the lifetime
of the building. Yet we can still refer to the height of the building, and to
its other dimensions, even if we have not measured these quantities and even if
they are unknown to some or all of us.
Here are some more questions, just for fun. What do we mean by the height of
the building? Before the building is built, can we talk about its height? Can
the height be taken to be zero? When the building is being built, is the height
of the building equal to the height of its walls as they are being put up? If
the building has a basement or a foundation, do we say the height of the
building is negative or is it undefined while the basement is being dug, or the
foundations being built? When the building is being demolished, does it have a
height? What is it?
What do we mean by height? Better yet, we can speak of the height of a
building whenever we can say what it represents (means) and/or how we might
measure it. This permits us to speak of the current height, the planned or
intended height, the past height, the future height. Is the height of a
demolished building zero, or undefined? Is the planned height of a building
equal to its actual height before construction, during construction, during its
use or during demolition? A definition or identification of the height we want
to speak about, is needed.
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: The mathematical usage of words
Next Topic: What
is a Variable:
| |
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
|