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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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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 ]
5 Numbers Versus Quantities
When you ask how tall I am, you may get the answer: 5 feet and 10 inches or
1.75 meters. The answer in either of its forms involves both numbers and units.
A number times a unit of measurement gives you a quantity. Quantities can be
added together: 5 feet plus 10 inches is 5[5/6] feet.
To further understand the difference between numbers and quantities, you may
ask how many pennies (or cents) I have in my pocket. The answer could be the
number 10. For the same pocket, if you asked how much money I had in it, the
answer would be the quantity 10 cents or even 0.10 dollars, a tenth of a dollar.
Numbers are given by counts - whole numbers, proper and improper fractions,
decimal numbers. Quantities are given by a count (a whole number or fraction)
times a unit of measurement. Any object that can be counted can serve as a unit
of measurement. Examples of units of measurement are: meter, foot, $ or dollar,
square foot, square meter, second, hour, meters per second, kilometers per hour,
dollars per hour, miles per hour and so on.
Numbers include no units. You get a number when you ask how many units there
are, and you have specified the unit. You get a quantity when you ask how much
there is. Saying a length is given by the number 5 is meaningless, if no units
of measurement are given. Saying a length is 5 raises the question 5 what?
The number 5 may give the number of units of length in a distance. Writing
this number by itself does not say what the unit of length might be. Some
information, the unit, is missing. So I repeat, in answering questions demanding
how much, we need to give a unit of measurement as well as a number. People
should not have to guess your unit of measurement when you speak. A length may
be given by 5 miles (or 8 kilometers). Of course, if we are asked how many miles
(or kilometers) there are in the length concerned, the number 5 (or 8) is
expected because the unit was specified. When you are asked how many people
there are in a room, you may respond with a pure number like 7 or 10. The unit
of measurement can be worded or written as person or persons.
In measurement and counting, a single unit of measurement, a fraction of one
or several units, may appear. For instance, a length of time may involve 1 hour
or 12.5 hours. Notice the addition of the letter s to the unit hour here when
fractions or more than one unit appears. In mathematics, we choose to ignore the
difference in spelling between the singular and the plural. If we insisted on
using the singular form, we would have to write 12.5 hours = 12.5 ×1 hour.
The latter gives the exact meaning of 12.5 hours. In writing units in
calculations, we may and will change their spelling (or abbreviations) according
to the rules of grammar. The plural and singular forms of each unit are declared
to be equal or interchangeable. Each is allowed to replace the other. Which one
sounds the most appropriate will be written in our formulas and calculations.
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-Changing or
Converting Units for Quantities.
Next Topic: What
is a Variable:
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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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