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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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.

Area Development of Multiplication
Rule for Polynomials

The foregoing area perspective of the distributive law points to a quick method for introducing the multiplication method for polynomials in one and if you like several variables.  This development also teaches the column method for addition of polynomials.

Area View of Product of Polynomials

Use rectangles to expand the product 

P = (50+ 6b +4b²)(8+ 6 b + 4b²+ 10b³)

where b is a whole number

Solution:  Form intermediate size rectangles 

and compute the number of subrectangles in each. The latter operation first gives

and then this 

The foregoing could have been done in one step. Now instead of add the intermediate rows by rows, we will add them along the diagonals in the powers of b are identical. 

If we have made no mistakes, the foregoing gives the result P = (50+ 6b +4b²)(8+ 6 b + 4b²+ 10b³)

  =  300  + (300 + 48)b + (200 + 36 + 32)b²  + (500+24+24)b³  +  (60+16)b⁴  +  40b⁵ 

 = 300  + 348b + 268b²  + 548b³  +  76b⁴  +  40b⁵ 

Whether or not the powers of b increase or decrease in the  result is a cosmetic convention - some prefer one way, others the other, and some either. 

Our conclusion follows. The product

P = (50+ 6b +4b²)(8+ 6 b + 4b²+ 10b³)  = 300  + 348b + 268b²  + 548b³  +  76b⁴  +  40b⁵ 

That is computation of the left hand side for a given value of b gives the same result as computation of the right. 

Multiplication Table Approach - Stepping towards a column
multiplication method for polynomials

The above geometric approach suggests a table method:

which holds for real number b as well as whole numbers since the generalized distributive law holds for real numbers as well as whole numbers (why to come later). Here the rows and columns of the table come from the terms in the factors of the product to be computed, here (50+ 6b +4b²)(8+ 6 b + 4b²+ 10b³) while the entries inside list or tabulate the products of pairs of terms, one from each factor.

Column Method for Multiplication of Polynomials

Here we modify the table approach and dedicate a column to each power of b as follows

In this modified table approach, we compute all possible products as before, but align the products in each row according to their power of b. That makes addition and collecting like powers of b (with the aid of the distributive law) simpler. We add an extra row for the sum.

So the product P = 300  + 348b + 268b²  + 548b³  +  76b⁴  +  40b⁵   as before

In retrospect, the table (and column multiplication below) will be easier to do if we arrange the powers of b in ascending (that is, increasing) order along the top row.

and align the left column at the left of the page (or page margin), so that we may write from left to right. Here again all possible products appear, but aligned in columns. 

Finally, let us introduce or switch to the table method for multiplication. 

Column Method for Products of Polynomials

8   +      6 b +   4b²  +  10b³ 
50 +      6b   +  4b²             ×
300 + 300b + 200b² + 500b³                              (product of top row with 50)
             48b  +  36b²  +   24b³  +  60b⁴                 (product of first row with 6b)
                        32b²  +   24b³   + 16b⁴   + 40b⁵  (product of first row with 4b²)
^{------------------------------------------------------------------------ +}
300 + 348b + 268b²  + 548b³  +  76b⁴  +  40b^{5 
------------------------------------------------------------------------}

Note: The ascending order appears to work best as the position of the  intermediate products shifts to the right away from the left margin where the computation begins.  Do you see how each entry in each row of the column method corresponds to an entry in the rectangular approach and the two preceding tabular approaches to the computation of the product? All is a consequence of the distributive seen algebraically or geometrically visualized.

Addition and Subtraction of Polynomials

In the above calculation, we see how to add  polynomials

(300 + 300b + 200b² + 500b³  ) + (48b  +  36b²  +   24b³  +  60b⁴ ) + (32b²  +   24b³   + 16b⁴   + 40b⁵   ) 
 
via an addition methods in which like terms are aligned in columns as follows.

300 + 300b + 200b² + 500b³                              
             48b  +  36b²  +   24b³  +  60b⁴                 
                        32b²  +   24b³   + 16b⁴   + 40b^{5  
------------------------------------------------------------------------ +}
300 + 348b + 268b²  + 548b³  +  76b⁴  +  40b^{5 
------------------------------------------------------------------------}

So one pedagogical methods is to developed the column method for multiplication first, and then introduce column methods for addition and subtraction of polynomials, second.

Multiplication and Addition Operations with Decimals

The case where the variable b is replaced by the number 10 leads to a justification of the column method for multiplication of whole numbers using their decimal representation with powers of 10 written in decreasing order rather than increasing, Some conversion need to be considered. See Distributive Law Consequences

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

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