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)  |> 
|      | |     |
   \             /   
\    =   /

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

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.

Visual Aids and Column Multiplication Methods

The association of products of whole numbers with counting subrectanglar divisions of a larger rectangle leads to visual aids for developing and applying the generalized distributive law for whole numbers, fractions, proper or not, and nonnegative real numbers in general. 

While the justifications here are only for positive real numbers, the calculations holds for real numbers.

While the principle of permanence of algebraic form or patterns was not a valid logical principle in the development or proof of the properties of growing sets of numbers from natural numbers to real and complex numbers,  in education the accidentally permanence of algebraic form can be a pedagogical tool in the development of algebraic skills and concepts.  Assuming that counting by grouping and the measure of perimeters, areas and volumes is independent of how counted or calculated. That provides a quick logical base for algebraic reasoning in the case of positive quantities, those identifiable with counts or measures. All the foregoing may lead to a logical development of algebra in which justifications are given for calculations involving positive quantities while students are informed that justification for general case involving positive, zero and negative quantities is a subject for further advanced study.

Students or teachers insist on the justification for general case, that is real numbers instead of only positive numbers, can develop proofs that apply mathematical induction and the distributive law, pattern or axiom for real numbers. Go to the site Number theory areas to learn more when you have time to spare

Area Development of Distributive Law
for positive numbers

First Example (Alternative to Foil Method)

How do we express a product 

NM = (a+b)(c+d) 

as a expression of the terms a, b, c and d?

Solution:  The number NM gives the number of subrectangles in the blue rectangle below.

The BLUE rectangle can be divided into  4 intermediate size subrectangles

Each intermediate rectangle labelled I to VI contains a number of the NM subrectangles we are counting. Each of the NM subrectangles we are counting belongs to one and only one of the intermediate size rectangles. See below.

Associated Column Multiplication Method

We have use the product rule for counting subrectangles to find the number of subrectangles of the total MN in each intermediate subrectangle. The intermediate size rectangles lead to four groups of subrectangles with counts ac, ad, bc and bd we can be add to obtain the total number MN. 

So  MN= (a+b)(c+d) = ab+ad + bc + bd.

We may introduce a column multiplication method to obtain the product 

 c + d
 a + b                  _x 
ac + ad                   = product of first row with a
bc + bd              ₊   = product of first row with b
ab + ad + bc + bd   =  (a+b)(c+d)

Here ab+ad + bc + bd  and   (a+b)(c+d) give two different ways to compute a single number, the number of subrectangles MN.  The equality of two different ways to compute a single number gives many formulas in mathematics. 

Second Example - Extension to more terms in factors

How do we express a product 

NM = (a+b+c)(e+f) 

as a expression of the terms a to f giving each factor.

Solution:  The number NM gives the number of subrectangles in the blue rectangle below.

The BLUE rectangle can be divided into  6 intermediate size subrectangles

Each intermediate rectangle labelled I to VI contains a number of the NM subrectangles we are counting. Each of the NM subrectangles we are counting belongs to one and only one of the intermediate size rectangles. See below.

We have use the product rule for counting subrectangles to find the number of subrectangles of the total MN in each intermediate subrectangle. The intermediate size rectangles lead to six groups of subrectangles with counts ea, eb, ec, fa, fb and fe we can be added in any order to obtain the total number MN. 

So  MN= (a+b+c)(e+f) = ea + eb + ec+ fa+ fb +fc 

We may introduce a column multiplication method to obtain the above product

Remark 1: The foregoing visual or geometric derivation the generalized distributive law  holds for non-negative rational and irrational numbers a to f with unit length in place of the word rows and columns if we derive and then use the additive properties of area - the area of a rectangle equals the sum of areas of a set of subrectangles covering it - subrectangles which intersect only at their edges. Details will be given later. 

Column Methods for Multiplication for the extension

 We may replace the rectangles above by multiplication tables in which the terms in the factors provide the initial entries in rows and columns. 

Further table entries are obtained via products. The foregoing can be tabulated as a column method for multiplication:

 a + b + c
 e + f                           _x 
ea + eb + ec                 = product of first row with e
fa + fb + fc                   ₊   = product of first row with f
ea + eb + ec+ fa+ fb +fc  =   (e+f)(a+b+c) or (a+b+c)(e+f) when multiplication
                                       is commutative

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

www.whyslopes.com
[Top of this Page] [Site Exit] [ Back ] [ Up ] [ Next ]
[Comments, Reactions, Feedback]
: Favourite Sites:  BBC News  and mathematics portion of  English National Curriculum  

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, a 1983 McGill. Ph. D. in mathematics
All Rights Reserved.