www.whyslopes.com:  Parents:  Help Your Child/ Teen Learn
Appetizers and Lessons for Mathematics and Reason  (Français)
Online tutor-teacher  seminars available

Online Volumes (Book Orders)
1,  Elements of Reason. (1996)
1A. Pattern Based Reason  (1995)
1B. Math Curriculum Notes (1996)
2. Three Skills for Algebra  (1995)
3.-Why Slopes & More Math  - (1995)

Read chapters 1-5, 8-12, 14, 16 & 17 
in  Three Skills for Algebra 

More Site Areas 
1.  Solving Linear Equations  (04-2005)
2.  Fractns, Rates Proportns, Units  (2006)
3.  Algebra, Odds & Ends, HS level-(2001)
4.  Euclidean Geometry  & Cmplx NOs-new 
5.  Analytic Geometry/Functions (2006)
6.  Number Theory. (2006-7)
7.  Complex Numbers (2001)
8.  Calculus Introduction (2005)
More Site Areas 
99   Real  Analysis (1995)
10. Secondary IV(?) maths (2006-7)
11. Math Education Essays  (2006-7)
12. LaTeX2HotEqn: (2004)
13. Electric Circuits Etc  (2007)
14. Quebec Math Ed (2004 -8))
15  Prequel to the How-TOs (06-2008)
16-Virtual-Classroom (Take a peek)
Tutor -Teacher-How TOs-(08- 2008)
1. Arithmetic Reference
2. Algebra 
3. More Algebra 
4. Geometry  
5. More Geometry
6. Calculus
7. Logics in Maths
Site Tour:  Click here,  here,  herehere,  here,   herehere and  here. Site Search Page


Employ an online or offline tutor at your own risk from 

AU:  tutorfinder.com.au
CDN :  findatutor.ca 
CDN: .i-tutor.ca
CDN: Montreal Tutors
NZ:   findatutor.co.nz
UK:   tutorhunt.com 
UK:  tutors4me.co.uk
USA:  wiziq.com
USA: ziizoo.com

YOU are better than YOU think. Show yourself  how:

      |      
//  _   _ \\
/\             /\
  <|  (o)   (o)   |> 
 \     | |      / 

 For better work & study skills, read logic chapters 1 to 5  in  Three Skills for Algebra. Sooner is better. Good luck.

 -/[]\- 
||
   / \_ 
 ||||||||||||||||||||||||||||


 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

Do not leave here without it -  Logic mastery  will develops critical thinking, improve reading and writing, and give a firmer base for work and studies at many levels. Good luck.

   |      
//  _   _ \\
/\             /\
<|   (o)   (o)  |> 
     | |     |
   \             /   
\    =   /

Caution: Site advice is approximately correct, for some circumstances, not all. Site How-TOs are logically developed, but not tried and tested. That leaves room for thought and refinement..

 -/[]\- 
||
  _ / \     
 ||||||||||||||||||||||||||||
 

After logic  (a) continue reading Three Skills for Algebra, chapters 8 to 14  and do so alongside site area on solving linear2007 Equations ; or (b) see this calculus starter lesson and Volume 3, Why Slopes  & More Math, chapters 2 to 6;


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.


Explore collaborative whiteboards from groupboardtwiddla  or scriblink.


Chapter 12
Shorthand Usage Guide

Previous Section: An Exercise


7  The Letters M, x, y and z

On calculators, a button marked M often refers to a memory location or the last number stored in that location. A number can be stored in and recalled from the memory location M. We may use the current value of M in calculations. For instance, the number described and given by the expression 5*M+4 can be computed on a calculator. The result obtained depends on the value saved for M. The expression 5*M+4 describes the calculation: multiply the value given or saved for M by the number 5 and then add 4.

We often meet the letters x, y and z in algebra. When you meet them, think of three boxes or memory spots, one for each. We can now speak and talk about the numbers (or quantity) in the first, second and third box, whatever they might be. To write about these numbers briefly, without too many words, we use the shorthand x, y and z. The letter x is our name or shorthand for the number in the first box. Similarly, the letters y and z become shorthand for the contents of the second and third boxes, respectively. The shorthand expression x ·y is now shorthand for the action: multiple the number in the first box by or with the number in the second box. The shorthand expression x·y describes a calculation which could be done if we knew what numbers or quantities were stored in the x and y boxes.

More generally, suppose the physical meanings or connotations of one or more letters are not given. Here we can imagine there are several containers labeled with these letters. Then we refer to the content of each box, a number or quantity, by the box label. The number q now refers to the number or quantity in the box or container labeled q - the latter also refers to the box itself.


The initial convention as begun in the 12-th century A.D. was to use vowels to represent unknown numbers and consonants to represent given ones. This stems from the convention in Arabic or Hebrew of writing only the consonants and then finding the vowels from the context of the words in which they appear.

 

Chapter Sections:  12 Symbols & Pronouns ] 12 Pronouns ] 12 Shorthand Usage Guide ] 12 Pronouns in Mathematics ] 12  Big and Small Letters ] 12  Subscripts Etc ] 12 An Exercise ] [ 12 Symbols & Numbers ] 12  Offspring Naming Conventions ] 12  A Review & Answers to Exercise ]

Next Chapter: 13 What's Next

 



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

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 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



Site Tour:  Click here,  here,  here,  
here
,   hereherehere and here.

On leaving, if your next stop would
 be an internet search, do it here.

Back ] Area Intro ] Up ] Next ] [Top of this Page]  
Road Safety Message  Do not walk on a road with your back to the traffic. 
  Favourite SitesBBC News  and mathematics portion of  English National Curriculum  
[Site Author Contact Form]  If you find an error, please cut and paste its 
web page location (URL) into its report.

All trademarks and copyrights on this page are owned by their respective owners.
Copyright to comments & contributions are owned by the Poster. 
The Rest © 1995 onward by site author,   Alan Selby (
email) All Rights Reserved. 

www.whyslopes.com