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.

Decimal Addition 

In our discussion of decimal methods below, we will need the property that 

 a 10^m + b 10^m =  (a+b) 10^m

for each whole number  m > =0

Counting Sub-Rectangles and Unit Squares

If a rectangle is divided into N = 4  columns and M = 3 rows into subrectangles, there will be Nx M = 3 x 4 = 12 subrectangles.

Begin Digression: If a rectangle is divided into N = 4  columns and M = 3 rows into unit squares, there will be N x M = 3 x 4 = 12 unit squares

	4 columns
3 rows

Counting the number unit squares needed to cover a rectangle whose sides are integral multiples of a unit length leads to the first notion of area - how unit squares are needed to cover the larger rectangle. Later one fractions of units squares may be included in defining the area of a rectangle with respect to a unit length.  Saying how to compute a number defines it. End of Digression

In general, division of a rectangle into N columns and M rows leads to NxM subrectangles. The same number results if we divide the rectangle into M columns and N rows. That follow from a reflection argument or the commutativity of multiplication for whole numbers.

But the columns need not have the same width and the rows need not have the same heights. An example follow.

Here there are 5 x 6 = 6 x 5 (green) subrectangles. Our ability to vary the dimension of rows and columns means the counted subrectangles do not have to be isometric - their dimensions (with and length) may vary.  We are only interested in how many. 

Column Method for Decimal Multiplication - An Example

The number  23  = 2 x 10 + 3 and the number 35 = 3 x 10 + 5. We suppose that these numbers give the number of rows and columns respectively in a large rectangle (the four coloured subrectangles) below. 

The number of subrectangles of each coloured subrectangle is as follows

The number of subrectangles in each row is indicated next

Now consider the column method for multiplication:

   35
x 23
105 = 15 + 3 x 30 = number of rectangles in first row
700 =  20 x 5 + 20 x 30 = number of rectangles in second row
805

The entries in this column method for multiplication correspond to 

We will speak more about the link between the distributive law and column methods for multiplication of decimal and polynomials later.

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Number Theory

Start of Number Theory

Origins of Counting or Tallying
Adding Wholes
Multipling Wholes
Distributive Law  Preamble
Distributive Law for Wholes
Consequences
More Consequences
What is a Fraction
Compound Fractions

Number Theory
Continued

Decimal Place Value
Place Value Reinforcement
Comparison Method
Addition Method
Subtraction Methods
Multiplication Methods
Division Methods
Remainder Arithmetic I
Primes & Composites
Primes Factorization Theorem
Primes & Composites
Prime Factorization Aids
Prime Factorization Examples
Counting  Whole No.  Factors
Arithmetic Videos
Square Roots  & Primes
Long Division Continued
Fractions & Decimals
Fractions as Decimals
1 = 0.999 Recurring
Infinite Decimals Expansion Arithmetic
Ratio of Simple Fractions
Ratio of Decimal Fractions
Unsigned Reals Numbers
Signed Coordinates
Plane Vectors
Horizontal Vectors
Adding Vector Multiplies
Adding Signed Numbers
Multiplying Signed Numbers
Distributive Law for Reals
Real Numbers Axioms
Remainder Arithmetic II

Related Site Pages:

• Complex Numbers
• Maps, Plans and Drawings

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

Food for thought: 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..

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