5. The Distributive Law for Whole Numbers
A consequence of the product law for counting equi-sized groups, and how
we add of counts.
Now
A units + B units = (A+B) units
Now suppose each unit is a distinct group of N elements. Then the A units
and the B units divide all elements present into two disjoint sets. The
objective is to count all, or more precisely to show we can count all via
two different groupings. Previous assumptions that counts or tallies are
independent of how a set is counted implies the equality of two different
ways of counting.
Now
-
in A units there are A × N elements,
-
in B units there are B × N elements, and
-
in (A+B)) units there are (A+B) × N elements,
The latter gives the total number of elements present. But the total
number is also given by the number in the A units plus the number in the B
units. Earlier on, we assumed that the count of a set of elements is
independent of how counted. Whence
A × N + B × N = (A+B) × N
due to the equality of two different ways count. We can tally the
elements in A units plus the elements in B units, or we can tally in the
elements in (A+B) units. The results should be equal.
Parallels in Daily Life - Origin of the above ideas
The above counting principle and the distributive law as is or extended
appears in many forms in daily life. For instance
12 apples + 14 apples = (12 + 14) apples = 26 apples.
Now if each apple correspond to or costs 5 cents, we say each apply
corresponds to set of 5 pennies. So we replace each apple by 5 pennies
and conclude that
12 × (5 pennies) + 14 × (5 pennies) = (12+ 14) × 5 pennies
Hence 12 × 5+14 × 5 tally marks (or ones) equals (12+14) × 5 tally marks
or ones in the sense that tallying the pennies in two different ways will
give the same decimal number for their count.
The teacher who gets students to write
3 N + 4 N = 7 N
via analogies
3 apples + 4 apples = (3+4) apples
is almost correct. The teacher is missing the following step: If
each apple is cut into N pieces, each apple corresponds to a set of N,
and the total number of apple pieces may be counted with the aid of the
product rule in two different ways. The ideas above provide the
missing step. However, for the pedagogical aim of developing a counting
skill, the apple analogy without the missing steps may be more
effective in that the including the missing step might overwhelm many
students - Ouch!
Polynomials in Powers of Ten
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Teachers & Tutors: Site pages offer better or best practices for providing skills -
simpler than expected & comprehensive but for exercises. For your charges, your duty is to study them alone or in
groups and develop skill building exercises & activities to share. Start now. The effort here is the best I can do.
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
with local curriculum control. Study how to include site content - its skill development how-TOs and innovations
into present or future lesson plans - some reading required.
Road
Safety Messages and Questions: When and why should you face
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See too, the BBC-Belgium story Texting and
Driving - texting & the impossible test - the article links to a gruesome utube video on the subject
The Logic of Injustice:
How Texas sent
an innocent man to his death - The wrong Carlos. Some judgments are irreversible. Procescution: Where and when prosectors play to win rather than for
justice, guilt beyond a reasonable doubt goes unrespected due to prosecutors who putting winning
first, those innocence before the law may be convicted. Some procescutors offices in continuing to accuse after a pardon
due to reasonable doubt or innocent being shown, may sucessfully oppose compensaton for false convictions
by asserting a pardon individual is still under suspicion. Then the pardoned individual or the latter's estate
is not compensation for years or decade
of improper or false imprisonment, or for execution. Site chapters on Logic
and some in Pattern
Based Reason may slowly lead to greater precision in reading, applying and
writing laws.
May 2012, Composition Starting:
Pre-School and Primary Mathematics - Quantitative Skills, An
Intellectual View, Feedback Welcome:
The 8 Most Popular Site Inlinks
Parent Center: Help your child or teen
learn:
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
science courses, and in developing organizational skills,
gives boys and girls a head start. Good luck. At the other
extreme, more comprehensive than light, if your motto is
McCainian: drill, drill, drill then Toronto
mathematician and actor John Mighton's jump math organization has jump math
workbooks for at least grades 3 to 8 for at-home and in-school
use - training sessions for teachers available. Jump math has
been expanding to cover older students. Jump Math Samples: plus
Fractions for
Grades 3-4 & Grades 5-6 [Read] Free Resources grades 1 to 8
[unread - likely to be good]. and
Mathematics
Skills For Ages 3 to 14 - technical!
Skills with take
home value - A few ideas
Basic skills include
time-date-calendar Matters; money matters; map, plan and
scale diagram matters;counting, measuring and figuring;
decision making with logic and likelyhood; being careful and
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writing with precision.
Is your child able to add, subtract and multiply amounts
of money, work with fractions, work with clocks and calendars,
work with maps and plans, and measure length, weight-mass and
volume? Schools may promote your son or daughter without
providing basic skills in reading, writing and
arithmetic.
Arithmetic
and Number Theory Skills
Algebra
Starter Lessons
Geometry
- maps plans trigonometry vectors
More
Algebra
70
Calculus Starter Lessons
Calculus Lessons Elsewhere:
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How to Ace Calculus: Street Wise Guide - Mostly
Text.
-
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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