Column Multiplication Method
The association of products of whole numbers with
counting sub-rectangular 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.
The following animated example gives another area based example of how to
example a product of two factors, when each factor is a sum of positive
terms or lengths.
Problem: How do we express a product
NM = (a+b+c)(e+f+g)
as a expression of the terms a to g?
Solution: The number NM gives the area A of the blue
rectangle - first calculation thereof.
The BLUE rectangle can be divided into subrectangles
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a
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b
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c
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e
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ea
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eb
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ec
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f
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fa
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fb
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fc
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g
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ga
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gb
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gc
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Next sum all sub rectangle areas, row by row.
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a
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b
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c
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Row Sums
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e
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ea
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eb
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ec
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ea + eb + ec
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f
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fa
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fb
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fc
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fa + fb + fc
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g
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ga
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gb
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gc
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ga + gb + gc
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So then the area A = ea + eb + ec+ fa+ fb +fc +ga + gb + gc as well.
That gives the second calculation of area. Therefore
(a+b+c)(e+f) = ea + eb + ec+ fa+ fb +fc +ga + gb + gc
as we assume different way of a calculating an area give the same value.
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
We may replace the rectangles above by multiplication tables in
which the terms in the factors provide the initial entries in rows and
columns.
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a
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b
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c
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Row Sums
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e
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ea
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eb
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ec
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ea + eb + ec
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f
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fa
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fb
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fc
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fa + fb + fc
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g
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ga
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gb
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gc
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ga + gb + gc
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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 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)
as multiplication is commutative
Remark: Even though the justifications above 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 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
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Teachers & Tutors: Site pages offer better or best practices for providing skills -
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Others are welcome to refine or exceed it. Please do.
Secondary
Mathematics for Ages 11+, A Practical Approach for home-tutoring or -schooling, or for schools & colleges
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How Texas sent
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of improper or false imprisonment, or for execution. Site chapters on Logic
and some in Pattern
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writing laws.
May 2012, Composition Starting:
Pre-School and Primary Mathematics - Quantitative Skills, An
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Parent-friendly
Work Booklets for ages 3+ to 13 Use these or others to check
or build skills. Other booklets are available but these booklets
allow parents unsure of themselves in mathematics to help their
children. The selection acquired in Canada is published in the
USA. So it has a US orientation. In retrospect, the selection
shows parents what to check with the booklets or by other ways,
the choice is theirs. But in retrospect, the selection does not
cover integral and fractions liquid weights and measures - ask
the publishers to correct that! For ages 9 to 12 say, parents may
compensate by showing boys and girls how to use weights or mass,
and further measures in food preparation. Beyond that children
may be shown how to measure and calculate angles, lengths and
areas [proportional amounts too] directly or by using maps and
plans drawns to scale. Learning how to gather and measure all the
ingredients, pots and pans for a dish or a meal, along with
cleaning up sets the stage for like activities or experiments in
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McCainian: drill, drill, drill then Toronto
mathematician and actor John Mighton's jump math organization has jump math
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Fractions for
Grades 3-4 & Grades 5-6 [Read] Free Resources grades 1 to 8
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Mathematics
Skills For Ages 3 to 14 - technical!
Skills with take
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Basic skills include
time-date-calendar Matters; money matters; map, plan and
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Calculus Starter Lessons
Calculus Lessons Elsewhere:
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How to Ace Calculus: Street Wise Guide - Mostly
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Flash
Video for Calculus Phobics
They cover basic topics in ways likely to complement your
notes, your textbooks and site material. When Goldilocks
trespassed in the house of the three bears, she found three bowls
of porridge, two not to her liking, and one just right. Different
bears have different tastes. As invited guest here and elsewhere,
if one or more explanations is not to liking, try another. It may
be better or just right.
Unsolicited Advice
Learning to do and high marks if it comes to easy is often
deceptive - light rather than deep. For that reason, students
with learning difficulties determined not to let it get in their
way may go deeper and farther than those with none. High marks,
if the come easy, may be deceptive - provide a too light and not
a deep mastery. That could have been your problem in secondary
school, one that leads to comprehension shock or difficulties in
calculus and more generally in the first year of college. Bon
Appetite.
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