Distributive Laws and Multiplication Methods
Distributive Laws and Multiplication Methods from Area Calculation
The division of a rectangle into non-overlapping sub-rectangles (apart from their borders) gives two different way to compute the former areas: directly from width times length; and indirectly as the sum of areas of the sub-rectangles. The assumed equality of the two different ways implies the distributive law for products of sums, where the sums involve positive terms.
Distributive Law, Geometrically implied. |
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a(b+c) = ab +ac as the area of the largest rectangle can be computed in two different ways, directly or as the sum of the areas ab and bc of the sub-rectangles. |
The next problem indicates the foil method for calculating (a+b)(c+d) is also a consequence of the equality of two different methods for calculating the area of a rectangle with sides of length a+b and c+d respectively.
Problem: 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.
| c columns | d columns | |
| a rows | Blue rectangle | |
| b row | ||
The BLUE rectangle can be divided into 4 intermediate size subrectangles
| c columns | d columns | |
| a rows | I | II |
| b row | III | IV |
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.
| c columns | d columns | |
| a rows | ac | ad |
| b row | bc | bd |
Thus
(a+b)(c+d) = ac + ad + bc +ad
upon addition by rows.
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.
| c columns | d columns | No in Each "Row" |
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| a rows | ac | ad | ac +ad |
| b row | bc | bd | bc+ bd |
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.
This geometric view of the distributive or general distributive laws that can
be used at many levels in high school mathematics.
Teachers: Column methods for addition and subtraction come essentially for free as part of the development of the column method for multiplication. The foregoing operations with the distributive law and polynomials are only valid, strictly speaking, for positive terms and variables. That being said, once the students have learnt how, I tell them to assume the distributive law and operations hold for all real numbers as well, and emphasize that I have only given them a partial derivation justification of the process. At this point in their development, students want to learn mathematics with an economy of thought and worry. So assumption are more to their liking than proofs, a liking to be reversed, we hope, in the logic part of this course.
Decimals and Polynomials (optional topic)
Multiplication, Addition and Subtraction of Decimals in expanded form can be provide examples of operations on polynomials in powers of 10 with the restriction that coefficients greater than 9 are also expressed as polynomials in power of 10, a conversion process leads in the end to the result in standard form, that is a polynomial in powers of ten with coefficients limited to the digits 0 to 9. To learn more see the discussion of Decimal Place Value and its consequences in the following webpages
Decimal Place Value
Comparison Method
Addition Method
Subtraction Methods
Multiplication Methods
Division Methods (Long)in the site section on Number Theory.
Another Example:
| Analytic Geometry Polynomials & Functions etc. |
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Extras |