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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

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